The motion of relativistic strings in curved space-times

This paper concerns the motion of a relativistic string in a curved space-time. As a general framework, we first analyze relativistic string equations, i.e., the basic equations for the motion of a one-dimensional extended object in a curved enveloping space-time $(\mathscr{N}, \tilde g)$, which is a general Lorentzian manifold, and then investigate some interesting properties enjoyed by these equations. Based on this, under suitable assumptions we prove the global existence of smooth solutions of the Cauchy problem for relativistic string equations in the curved space-time $(\mathscr{N}, \tilde g)$. In particular, we consider the motion of a relativistic string in the Ori's space-time, and give a sufficient and necessary condition guaranteeing the global existence of smooth solutions of the Cauchy problem for relativistic string equations in the Ori's space-time.